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Gradient of a Scalar Field

As we known that the value of a scalar function is constant at a fixed point in space, so the

gradient of a scalar function θ is determined as

 

grd θ = i

 

where ∆ is an operator ‘del’.

 

Physical Significance of Gradient

 

A scalar field may be represented by a series of level surfaces each having a constant value of scalar point function θ. Examples of these surfaces is isothermal, equidensity and equipotential surfaces. The θ changes by a constant value as we move from one surface to another. These surfaces are known as Gaussian surfaces. Since θ is a single valued function, the two surfaces cannot intersect each other. Now let the two such surfaces are very close   together, be  epresented by two scalar point functions  and (θ + d θ ). Let r and (r + d θ) be the position vectors of points A and Bon the surfaces θ and (θ + d θ) respectively with respect to an origin 0 as shown in Figure 7.1. Clearly the vector AB will be dr. Let the minimum distance between the two surfaces dn be in the direction of unit normal vector n at A.

 

From Figure 7.1 it is clear

Dn  =dr cos  θ

 

= | n | dr | cos θ =n .dr

 

Dϕ= ∂ϕ/  dn  =∂ϕ/dn n .dr

 

Since the continuous scalar function defining the level surfaces (Gaussian surfaces) has a value θ at point A (x, y, z) and (θ + dθ) at point (x + dx, y + dy, z + dz), we have

 

Dϕ = ∂ ϕ/dx dx +dϕ/∂y + ∂ϕ/∂x dz

 

=( I ∂ϕ/∂x +j ∂ϕ/vy +k ∂ϕ/∂z) .(idx +jdy +kdz)

 

=∆ ϕ. dr

 

From equations (1) and (2), equating the values of d θ,

 

We obtain ∆θ .dr =∆ ϕ= ∂ϕ/ ∂n n .dr

 

As dr is any arbitrary vector, we have

 

∆ϕ =∂ϕ/∂ n

 

Grad ϕ = ∂ ϕ/∂n n

But  θ / n   n  is the maximum rate of increase for a scalar function θ . Therefore, the gradient an of a scalar field at any point is a vector field, the magnitude of which is equal to the maximum  rate of increase of θ  at that point and the direction of it is same as that of normal to the level surface at that point.